CROATICA CHEMICA ACTA
CCACAA, ISSN 0011-1643, e-ISSN 1334-417X
Croat. Chem. Acta 87 (3) (2014) 287–290.
http://dx.doi.org/10.5562/cca2506
Original Scientific Article
Theory of Anodic Stripping Square Wave Voltammetry on
Spherical Mercury Electrodes
Šebojka Komorsky-Lovrić and Milivoj Lovrić*
Department of marine and environmental research, “Ruđer Bošković” Institute, 10000 Zagreb, Croatia
RECEIVED JUNE 6, 2014; REVISED OCTOBER 3, 2014; ACCEPTED NOVEMBER 18, 2014
Abstract. Relationships between dimensionless anodic stripping square-wave voltammetric net peak currents and the dimensionless inverse electrode radius are curves with two asymptotes that depend on the
duration of accumulation. The theory applies to reversible reduction of amalgam forming ions on stationary macro and micro spherical and hemispherical mercury electrodes.
Keywords: square wave voltammetry, spherical electrodes, theory, microelectrode
INTRODUCTION
The contamination of aquatic ecosystem by metal ions
can be monitored by networks of submersible probes.1–5
For trace metal analysis the technique that is very well
suited is stripping voltammetry in combination with
microelectrodes.6–10 These electrodes are characterized
by high mass transport density, small double-layer capacitance, small ohmic drop and steady-state current
response in unstirred solutions.11–15 Thus, in the preconcentration step of stripping technique no agitation of
solution is necessary.2,4,9,11,12 The concentration of
amalgam forming ions can be measured by anodic stripping voltammetry on mercury microelectrodes.16,17 The
application of square-wave voltammetry in the stripping
phase is particularly efficient.18 The theory of this technique is well developed for the stationary planar electrode covered by thin mercury film,19 but not for spherical electrodes because of the need to consider the diffusion within the mercury drop.20–22 This problem was
investigated for the cathodic reaction, without the accumulation, in our previous paper.23 In this communication the dependence of anodic stripping square-wave
voltammetric net peak current and peak potential on the
electrode radius and frequency is investigated theoretically for the reversible reduction of amalgam forming
ions.
THE MODEL
A reversible electrode reaction on mercury drop electrode is considered:
Ox n   ne  Red(Hg)
(1)
t  0; r  rm :
(2)
0  r  rm :
*
cOx  cOx
, cRed  0
cOx  0, cRed  0
t  0; r   :
cOx  c*Ox
r  rm : (cOx )r rm  (cRed )r rm exp(φ)
(3)
(4)
(5)
nF
( E  E )
RT
(6)
I
 c 
D  Ox   
nFS
 r r rm
(7)
I
 c 
D  Red   
nFS
 r r rm
(8)
φ
r  0:
 cRed 
 r   0

r 0
(9)
The meanings of symbols are the following: cOx
and cRed are concentrations of ions in the electrolyte and
*
atoms in the mercury, respectively, cOx
is the bulk concentration of ions, E is electrode potential, E° is stand-
* Author to whom correspondence should be addressed. (E-mail: mlovric@irb.hr)
288
Š. Komorsky-Lovrić and M. Lovrić., Theory of Anodic Stripping Square Wave Voltammetry
ard potential of electrode reaction (1), D is a common
diffusion coefficient, I is a current, n is a number of
electrons, F is Faraday constant, S is the electrode surface area and rm is the radius of mercury electrode.
The mass transport is calculated by the Feldberg
approximation.24 The simulation procedure is described
in the previous paper.23 The results are reported as di*
mensionless current   I ( nFScOx
) 1 ( Df ) 0.5 . In square
wave voltammetry the current is sampled at the end of
each pulse and the difference between two subsequent
samples is called the net response: ΔΦ = Φf – Φb. The
forward, oxidative (Φf) and the backward, reductive
(Φb) components of the net response are also reported as
a function of the potential of staircase ramp. The responses depend on the pulse amplitude Esw, the potential
step dE, the inverse value of dimensionless electrode
radius ρ  rm1 ( D / f ) 0.5 and the dimensionless parameter kacc that defines the relationship between the accumulation time and frequency: kacc = 50 tacc f. In the simulation each pulse is divided into 25 time increments, so
that Δt = (50 f)–1 and kacc = tacc / Δt. The number of time
increments determines the precision of simulation.25 The
calculations were simplified by assuming that both ions
and atoms have equal diffusion coefficients.
RESULTS AND DISCUSSION
In anodic stripping square wave voltammetry the
starting potential is much lower than the standard
potential, the scan direction is positive and the
stripping scan is preceded by the accumulation period during which the electrode is charged to the
starting potential. In our calculations this process
was simulated by the diffusion controlled reduction
at constant potential during several seconds. Two
examples of stripping voltammograms are shown in
Figure 1. If ρ = 0.03 the net dimensionless peak
current is 2.00 and peak potential is –0.006 V vs.
E°. The maximum and minimum of the oxidative
and reductive components are 1.13 and –0.86, respectively, and the potentials of these extremes are
–0.004 V and –0.008 V, respectively. For ten times
higher value of ρ the net peak current is 6.61 times
bigger than the same current in Figure 1a. The maximum of oxidative component is enhanced 7.33
times and the minimum of reductive component is
5.77 times deeper comparing to the extremes of
respective components in Figure 1a. The peak potential in Figure 1b is –0.074 V vs. E°, which is 68
mV lower than in Figure 1a.
The influence of the parameter kacc on the concentration of amalgam at the end of accumulation period
can be seen in Figure 2, for r / rm ≤ 1. If the radius of
electrode is 100 μm and the diffusion coefficient is
9×10–6 cm2 s–1, these profiles correspond to the real
parameters tacc / s = 0.4 (1), 2 (2) and 10 (3). The
corresponding concentrations of reactant are shown for
r / rm ≥ 1 The gradients of the later curves at the electrode surface can be defined as functions of the bulk
concentration of reactant and the distance  that is called
the diffusion layer thickness:
*
cOx
 cOx 

 r 
δ

r rm
(10)
This distance depends on the accumulation time
and electrode radius.26 It is characterized by two limiting values: lim rm  δ  πDtacc and limtacc  δ  rm .
The second limit corresponds to the steady-state
conditions. Figure 3 shows the dependence of dimensionless thickness of diffusion layer on the accumulation parameter kacc. The points were calculated by the
simulation, using Eq. (10), and the line is theoretical
Figure 1. Anodic stripping square wave voltammograms of electrode reaction (1). Eacc = –0.300 V vs. E°, Est = Eacc,
Esw = 0.050 V, dE= 0.002 V, kacc = 1×104 and (a) ρ = 0.03; (b) ρ = 0.30.
Croat. Chem. Acta 87 (2014) 287.
289
Š. Komorsky-Lovrić and M. Lovrić., Theory of Anodic Stripping Square Wave Voltammetry
Figure 2. Relationship between relative concentrations of
metal ions and amalgam at the end of accumulation period and
the dimensionless distance from the center of spherical electrode. Eacc = −0.300 V vs. E°, ρ = 0.03 and (1) kacc = 2×103;
(2) kacc = 1×104; (3) kacc = 5×104.
relationship:
m πdkacc
δ

r m  πdkacc
(11)
Here, d = D Δt Δr–2 and m is the number of space
increments into which the electrode radius is divided
(rm = m Δr). The relationship (11) tends to the limit m =
149, which corresponds to ρ = 0.03, but the tendency is
rather slow for kacc > 105 and the ratio  / Δr becomes
higher than 95 % of this limit if kacc > 7.27×106.
Because of relatively stable mass transport, the
dependence of stripping peak current on the accumulation parameter can be approximated by the straight line
ΔΦp = 4.42×10–5 kacc + 2.37. This is shown in Figure 4.
Figure 5 shows the dependence of net peak current
Figure 3. Relationship between dimensionless diffusion layer
thickness and the accumulation parameter. All other data are
as in Figure 2.
Figure 4. Relationship between dimensionless net peak current and the accumulation parameter; ρ = 0.03 and all other
data are as in Figure 1.
and peak potential on the parameter ρ. The relationships
between p and ρ are curves that depend on the
parameter kacc. The straight lines in Figure 5a are
asymptotes that are defined by the following equations:
p = 35 ρ + 0.5, for ρ < 0.1, and p = 10 ρ + 4, for
ρ > 0.2, both for kacc = 2×103, p =63.5 ρ + 0.1, for
ρ < 0.1, and p = 22.5 ρ + 6.4, for ρ > 0.2, both for
kacc = 1×104, and p = 163.4 ρ for kacc = 5×104. The
conditions ρ < 0.1 and ρ > 0.2 mean that rm  10 D / f
and rm  5 D / f , respectively. These asymptotes
show the relationships between the real net peak currents on hemispherical microelectrodes, with the surface
area  2πrm2 , and the electrode radius and frequency.
For instance, if ρ > 0.2 and kacc = 1×104, this dependence is given by the equation:

*
I p  2πnFcOx
rm D 6.4rm f  22.5 D

(12)
Assuming that D = 9×10–6 cm2 s–1 and f = 100 Hz,
two terms in the brackets of Eq. (12) are equal if rm =
10.55 μm. The calculations show that the contribution
of transient (frequency dependent) component is
decreased as the electrode radius is diminished and the
duration of accumulation is increased. The steady-state
(frequency independent) stripping peak current calculated from Eq. (12) is given by the following equation:
*
I p  45πnFcOx
rm D .
Figure 5b shows that the peak potentials are practically independent of the parameter kacc. The straight
lines are the first approximations that are defined by the
equations Ep – E° = –0.30 ρ + 0.003 (V) and Ep – E° =
–0.13 ρ – 0.032 (V), respectively.
In these calculations the low solubility of metal
atoms in mercury is ignored and the theory applies to
ions in traces only.
Croat. Chem. Acta 87 (2014) 287.
290
Š. Komorsky-Lovrić and M. Lovrić., Theory of Anodic Stripping Square Wave Voltammetry
Figure 5. Dependence of anodic stripping square wave voltammetric (a) net peak current; and (b) peak potential on the dimensionless inverse electrode radius; (1) kacc = 2×103; (2) kacc = 1×104; and (3) kacc = 5×104. All other parameters are as in Figure 1.
CONCLUSION
Anodic stripping square-wave voltammetry on spherical
microelectrodes is characterized by near steady-state
conditions. The influence of spherical diffusion is
higher if the accumulation is longer and electrode radius
is smaller. Because of restricted diffusion field within
mercury drop, the maxima and minima of oxidative and
reductive components of the response do not vanish as
the radius is diminished. The real stripping peak current
depends linearly on the square root of frequency, with
the intercept that is equal to the steady-state current.
Acknowledgements. The financial support by the Croatian
Science Foundation in the frame of the project number IP-112013-2072 is gratefully acknowledged.
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